Chemical potential of a test hard sphere of variable size in hard-sphere fluid mixtures

Abstract

A detailed comparison between the Boublík-Mansoori-Carnahan-Starling-Leland (BMCSL)equation of state of hard-sphere mixtures is made with Molecular Dynamics (MD) simulations of the same compositions. The Labík and Smith simulation technique [S. Labík and W. R. Smith, Mol. Simul. \textbf{12}, 23-31 (1994)] was used to implement the Widom particle insertion method to calculate the excess chemical potential, $\beta {\mu }_0^{\text{ex}}$, of a test particle of variable diameter, ${\sigma }_0$, immersed in a hard-sphere fluid mixture with different compositions and values of the packing fraction, $\eta$. Use is made of the fact that the only polynomial representation of $\beta {\mu }_0^{\text{ex}}$ which is consistent with the limits ${\sigma }_0\to 0$ and ${\sigma }_0\to \infty$ has to be of the cubic form, i.e., $c_0(\eta )+{\overline{c}}_1(\eta ){\sigma }_0/M_1+{\overline{c}}_2(\eta )({\sigma }_0/M_1{)}^2+{\overline{c}}_3(\eta )({\sigma }_0/M_1{)}^3$, where $M_1$ is the first moment of the distribution. The first two coefficients, $c_0(\eta )$ and ${\overline{c}}_1(\eta )$, are known analytically, while ${\overline{c}}_2(\eta )$ and ${\overline{c}}_3(\eta )$ were obtained by fitting the MD data to this expression. This in turn provides a method to determine the excess free energy per particle, $\beta a^{\text{ex}}$, in terms of ${\overline{c}}_2$, ${\overline{c}}_3$, and the compressibility factor, $Z$. Very good agreement between the BMCSL formulas and the MD data is found for $\beta {\mu }_0^{\text{ex}}$, $Z$, and $\beta a^{\text{ex}}$ for binary mixtures and continuous particle size distributions with the top-hat analytic form. However, the BMCSL theory typically slightly underestimates the simulation values, especially for $Z$, differences which the Boublík-Carnahan-Starling-Kolafa formulas and an interpolation between two Percus-Yevick routes capture well in different ranges of the system parameter space.

11(+5) pages, 10 figures; Supplementary Material included; v2: slight change in title plus other minor changes